Theory seems to indicate that an offset should be included in count models if groups are observed for different lengths of time or if they have different levels of exposure.

Suppose we are observing kids playing basketball during open gym with groups A and B. Group A tires out after playing 20 minutes and leaves the court after scoring 2.5 goals each on average. Group B perseveres another 20 minutes and scores a total of 5 goals on average per student.

It is true that the two groups had different lengths (minutes) of duration or 'exposure' to playing time. People are telling me because of this difference I need to include minutes of playing time as an offset to account for this but I am not so sure.

In this instance there seem to be important differences in group A and B in terms of drive and ambition that should not be equated by accounting for minutes played or inclusion of an offset. Event success (scoring more goals) seems to drive time as much as time drives the event. Time seems endogenous (see p. 89), or confounded with the outcome or confounded with other factors like effort and motivation to play longer which drive the outcome. Should playing time in minutes be included as an offset for data generated from a scenario similar to this? How should this be modeled?

Or ......would it make more sense for the exposure (playing time) to be measured in hours instead of minutes? (effectively measuring goals per hour would indicate group B was superior to A vs a goals per minute comparison)

This is not a perfect story but this scenario parallels closely the outcome data and behavior I am challenged with modeling.

$\begingroup$Do you have any data (variables) that measure motivation or something like that, or that could serve as instruments?$\endgroup$
– gung♦Mar 19 '17 at 1:04

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$\begingroup$I think that is a good suggestion. Once I can determine if an offset makes sense in this case (does it make sense to use playing time as an exposure) I think that is the next question...is it endogenous...and if it is what do we do about it. I think your suggestion related to instruments is a very nice one for that scenario.$\endgroup$
– M. BogardMar 19 '17 at 17:42